#466957

order	512
nilpotency class	3
isoclinism	isoclinic class of #454382
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1262 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,16,17)(15,18,19,20)(21,22,25,27)(23,28,24,26);(6,9)(10,12)(14,17)(18,20)(21,23,27,28,25,24,22,26);(5,7,8,11)(6,10,9,12)(14,17)(15,19)(21,24,25,23)(22,26,27,28);(2,4)(7,11)(10,12)(13,15,16,19)(14,18,17,20)(23,24)(26,28)

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,c^2=ef,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320003, #72320228, #72320254
maximal normal subgroup of	#72320003, #72320228, #72320254
maximal quotient of	#72322635

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 374153);